The Epsilon Expansion Meets Semiclassics

We study the scaling dimension $\Delta_{\phi^n}$ of the operator $\phi^n$ where $\phi$ is the fundamental complex field of the $U(1)$ model at the Wilson-Fisher fixed point in $d=4-\varepsilon$. Even for a perturbatively small fixed point coupling $\lambda_*$, standard perturbation theory breaks down for sufficiently large $\lambda_*n$. Treating $\lambda_* n$ as fixed for small $\lambda_*$ we show that $\Delta_{\phi^n}$ can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in $$\Delta_{\phi^n}=\frac{1}{\lambda_*}\Delta_{-1}(\lambda_* n)+\Delta_{0}(\lambda_* n)+\lambda_* \Delta_{1}(\lambda_* n)+\ldots$$ We explicitly compute the first two orders in the expansion, $\Delta_{-1}(\lambda_* n)$ and $\Delta_{0}(\lambda_* n)$. The result, when expanded at small $\lambda_* n$, perfectly agrees with all available diagrammatic computations. The asymptotic at large $\lambda_* n$ reproduces instead the systematic large charge expansion, recently derived in CFT. Comparison with Monte Carlo simulations in $d=3$ is compatible with the obvious limitations of taking $\varepsilon=1$, but encouraging.